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Properties

Label 24-1813e12-1.1-c0e12-0-1
Degree $24$
Conductor $1.261\times 10^{39}$
Sign $1$
Analytic cond. $0.301064$
Root an. cond. $0.951212$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 2·8-s + 3·16-s − 6·32-s + 6·43-s − 64-s − 12·79-s + 12·107-s + 127-s − 6·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3·4-s − 2·8-s + 3·16-s − 6·32-s + 6·43-s − 64-s − 12·79-s + 12·107-s + 127-s − 6·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(7^{24} \cdot 37^{12}\)
Sign: $1$
Analytic conductor: \(0.301064\)
Root analytic conductor: \(0.951212\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 7^{24} \cdot 37^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.256990412\)
\(L(\frac12)\) \(\approx\) \(2.256990412\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
37 \( ( 1 - T^{2} + T^{4} )^{3} \)
good2 \( ( 1 - T^{2} + T^{4} )^{3}( 1 + T^{3} + T^{6} )^{2} \)
3 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
5 \( 1 - T^{12} + T^{24} \)
11 \( ( 1 - T^{6} + T^{12} )^{2} \)
13 \( 1 - T^{12} + T^{24} \)
17 \( 1 - T^{12} + T^{24} \)
19 \( 1 - T^{12} + T^{24} \)
23 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
29 \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \)
31 \( ( 1 - T^{4} + T^{8} )^{3} \)
41 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
43 \( ( 1 - T + T^{2} )^{6}( 1 - T^{2} + T^{4} )^{3} \)
47 \( ( 1 - T^{2} + T^{4} )^{6} \)
53 \( ( 1 + T^{2} )^{6}( 1 - T^{6} + T^{12} ) \)
59 \( 1 - T^{12} + T^{24} \)
61 \( 1 - T^{12} + T^{24} \)
67 \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \)
71 \( ( 1 - T^{3} + T^{6} )^{4} \)
73 \( ( 1 - T + T^{2} )^{6}( 1 + T + T^{2} )^{6} \)
79 \( ( 1 + T )^{12}( 1 - T^{6} + T^{12} ) \)
83 \( ( 1 - T^{6} + T^{12} )^{2} \)
89 \( 1 - T^{12} + T^{24} \)
97 \( ( 1 - T^{4} + T^{8} )^{3} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.95889718310821944252730481647, −2.95522343156498714974764263899, −2.93954166677501763141286384686, −2.93797595541624295585277024624, −2.79945590823809507595853193798, −2.76752082671652384611591420035, −2.65131972664061747066109529202, −2.52441129946525708601380048100, −2.38647137636919898993480455252, −2.28213066133972092965855462590, −2.15197390708898651856538124995, −2.12679054906007423511418522920, −2.07034920227494139951162893738, −1.94650792677968862383588073982, −1.84507058609830641218369445736, −1.84216508529130296730817781564, −1.58678680903540475991523524959, −1.57062007138022422796642629158, −1.51890180467044866030331174014, −1.27946744102393274068533907831, −1.03786305574000725485701793842, −1.00609520639536831618398149609, −0.819211214913130000970697026814, −0.72918723934826751251543598091, −0.45485629054885439468061372787, 0.45485629054885439468061372787, 0.72918723934826751251543598091, 0.819211214913130000970697026814, 1.00609520639536831618398149609, 1.03786305574000725485701793842, 1.27946744102393274068533907831, 1.51890180467044866030331174014, 1.57062007138022422796642629158, 1.58678680903540475991523524959, 1.84216508529130296730817781564, 1.84507058609830641218369445736, 1.94650792677968862383588073982, 2.07034920227494139951162893738, 2.12679054906007423511418522920, 2.15197390708898651856538124995, 2.28213066133972092965855462590, 2.38647137636919898993480455252, 2.52441129946525708601380048100, 2.65131972664061747066109529202, 2.76752082671652384611591420035, 2.79945590823809507595853193798, 2.93797595541624295585277024624, 2.93954166677501763141286384686, 2.95522343156498714974764263899, 2.95889718310821944252730481647

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.